Using the maximum principle (Theorem 8.1 [1]}), \(u\in \mathfrak {C}\) . In fact, we can let \( \varphi = (w-T_{max})_+\in H^{1}_0 ( \Omega )\) ,
\(C(\Omega )\lambda \Vert (w-T_{max})_+\Vert _1^2&\le & \lambda |(w-T_{max})_+|_1^2\nonumber \\&\le & \int _\Omega A(z(x),x )\nabla (w-T_{max})_+ \cdot \nabla (w-T_{max})_+\nonumber \\&=& \int _\Omega A(z(x),x )\nabla w \cdot \nabla (w-T_{max})_+= 0.\) 
